Department Mathematik



Ribbon transformations, Integrals, and
Triangular Decompositions

Yorck Sommerhäuser


We study the theory of integrals in Yetter-Drinfel'd Hopf algebras and use the results to determine the integrals of Hopf algebras with triangular decomposition.


The question whether it is possible to understand the algebra structure of deformed enveloping algebras in terms of the algebra structure of the three subalgebras appearing in the triangular decomposition was addressed in [31] (cf. also [32]). For this purpose, one can use two Hopf-algebraic constructions, the first one of these two being a very general one which might describe the structure of other Hopf algebras, too. In these constructions, two of the three components are not ordinary Hopf algebras but Hopf algebras in a certain twisted sense, that is, Hopf algebras in the category of Yetter-Drinfel'd modules. It is the purpose of this paper to determine the integrals of Hopf algebras that admit a triangular decomposition which is similar to the triangular decomposition of deformed enveloping algebras. For this purpose, we carry out a comprehensive investigation of the properties of integrals of Yetter-Drinfel'd Hopf algebras.

The article is organized as follows: In section 2, we recall some results of D. Fischman, S. Montgomery and H.-J. Schneider (cf. [5]) concerning the existence and uniqueness of integrals in Yetter-Drinfel'd Hopf algebras, as well as the definition of the integral character and the integral group element. It must be emphasized that these authors have established these results even under much more general hypotheses. In contrast to their methods, the proofs in section 2 are direct analogues to the proofs for ordinary Hopf algebras. These proofs also carry over directly to more general quasisymmetric categories as already observed earlier by V. Lyubashenko (cf. [16], [17]). As H.-J. Schneider has pointed out, these facts are also recalled in a recent preprint of Y. Doi (cf. [4]). It should be noted that important parts of these results have already been shown much earlier by D. Radford (cf. [23]). After this repetition, we study the properties of the modular functions and elements as well as the properties of integral character and the integral group element. In particular, we show that these are central elements in a certain sense.

In section 3, we lay the abstract foundations that make it easier to understand the nature of certain maps that arise naturally when considering Yetter-Drinfel'd Hopf algebras. We recall the notion of a monoidal transformation and contrast it with the related notion of a ribbon transformation. We exhibit examples for monoidal and ribbon transformations in the category of Yetter-Drinfel'd modules, and explain the effect of the action of such transformations on Yetter-Drinfel'd Hopf algebras. Two kinds of monoidal transformations, the modular transformations and the integral transformation, play the essential role in section 4 where we introduce the twisted Nakayama automorphisms and prove an analogue of Radford's formula for the fourth power of the antipode of a Hopf algebra in the Yetter-Drinfel'd case.

In section 5, we determine the integrals of Hopf algebras with triangular decomposition that arise from the above mentioned constructions. It turns out to be difficult to determine the integrals for the first construction, while the integrals of the second construction can be easily obtained afterwards. These results contain as a special case the formula for the integrals of the Drinfel'd double construction obtained by D. Radford (cf. [24]). In the last section 6, we illustrate the theory by considering the example of the Frobenius-Lusztig kernel of sl(2). The Frobenius-Lusztig kernels of deformed enveloping algebras were defined by G. Lusztig (cf. [14], [15]). Their present name was introduced by N. Andruskiewitsch and H.-J. Schneider (cf. [1]).

In this article, all vector spaces occurring are defined over a base field K. An excellent general reference for all topics discussed here is [19].